Circular motion — centripetal acceleration derivation

1.2.15 · Physics › Newton's Laws & Dynamics

WHAT / WHY / HOW

WHY exist karti hai? Newton ki 1st law kehti hai ki koi body straight line mein chalti rehti hai jab tak koi force na lage. Circle ek straight line nahi hai, isliye velocity vector ko constantly inward bend karna padta hai. Bending = acceleration. Koi inward acceleration nahi ⇒ object tangentially ud jayega.

WHAT direction? Hamesha velocity ke perpendicular, center ki taraf point karti hai. Kyunki yeh perpendicular hai, yeh $v$ ki direction badhalti hai lekin kabhi uski magnitude nahi (yeh koi work nahi karti, kyunki $a ⊥ v$ ).

HOW big? Yahi derivation neeche hai.

Derivation from first principles

Hum isse do tareekon se karte hain taaki tum isse scratch se rebuild kar sako.

Method 1 — Velocity triangle ki Geometry (similar triangles)

Figure — Circular motion — centripetal acceleration derivation

Ek object ko consider karo jo radius $r$ ke circle par constant speed $v$ se move kar raha hai.

Time $t_{1}$ par woh point A par hai; thodi der $Δ t$ baad woh point B par hai. Usne center par ek chhota angle $Δ θ$ sweep kiya hai.

Step 1 — Position triangle. Dono radius vectors $r_{A}$ aur $r_{B}$ ki equal length $r$ hai aur woh angle $Δ θ$ enclose karte hain. Chord AB ki length, small angle ke liye, $∣ AB ∣ \approx r Δ θ .$ Yeh step kyun? Arc length $= r Δ θ$ hai, aur small angle ke liye chord ≈ arc hota hai.

Step 2 — Velocity triangle. Velocity circle ke tangent hai, isliye $v_{A} ⊥ r_{A}$ aur $v_{B} ⊥ r_{B}$ . Dono vectors ko $90°$ rotate karne se unke beech ka angle nahi badalta — toh $v_{A}$ aur $v_{B}$ bhi wahi angle $Δ θ$ se separate hain. Dono ki length $v$ hai.

Yeh step kyun? Yahi key insight hai: velocity vectors ek aisa triangle banate hain jo position triangle ke similar hai (do equal sides, same enclosed angle).

Step 3 — Similar triangles. Velocity change ki magnitude $∣Δ v ∣$ ko chord ke against set karo: $\frac{∣Δ v ∣}{v} = \frac{∣ AB ∣}{r} = \frac{r Δ θ}{r} = Δ θ .$ Toh $∣Δ v ∣ = v Δ θ .$ Yeh step kyun? Similar triangles ke corresponding sides proportional hote hain. Dono triangles mein "short side over long side" ratio same hota hai.

Step 4 — Limit lo. Acceleration magnitude: $a_{c} = lim_{Δ t \to 0} \frac{∣Δ v ∣}{Δ t} = lim_{Δ t \to 0} \frac{v Δ θ}{Δ t} = v \frac{d θ}{d t} = v ω .$ Kyunki $v = ω r \Rightarrow ω = v / r$ : $a_{c} = v ω = \frac{v ^{2}}{r} = ω^{2} r$

Step 5 — Direction. Jaise $Δ t \to 0$ , $Δ θ \to 0$ aur $Δ v$ , $v$ ke perpendicular ho jaata hai, center ki taraf point karta hua. Isliye "centripetal."

Method 2 — Calculus (position vector)

Position ko yun define karo: $r (t) = r (cos ω t, sin ω t),$ constant $r$ , constant angular speed $ω$ .

Yeh form kyun? Radius $r$ ke circle par koi bhi point inhi coordinates rakhta hai; $ω t$ woh angle hai jo sweep hua hai.

Ek baar differentiate karo (velocity): $v = \dot{r} = r ω (- sin ω t, cos ω t), ∣ v ∣ = r ω = v .$

Dobara differentiate karo (acceleration): $a = \dot{v} = - r ω^{2} (cos ω t, sin ω t) = - ω^{2} r .$

Yeh step kyun? $cos$ ka second derivative $ω$ ke do factors aur ek minus sign le aata hai.

$- r$ factor dikhata hai ki $a$ radius vector ke opposite point karta hai, yaani seedha center ki taraf, magnitude ke saath $∣ a ∣ = ω^{2} r = \frac{v ^{2}}{r} . ✓$

Dono methods agree karte hain — yahi tumhara confidence check hai.

Worked examples

Forecast-then-Verify

Recall Pehle forecast karo, phir check karo

Q: Agar tum speed double kar do lekin radius same rakho, toh $a_{c}$ ka kya hoga? Padhne se pehle predict karo.

A: $a_{c} \propto v^{2}$ , toh $v$ double karne par $a_{c}$ chaar guna ho jaata hai (×4). Isliye high-speed corners itne dangerous hote hain: required centripetal force $v^{2}$ ke saath scale karta hai.

Common mistakes (Steel-man + fix)

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tum ek ball ko string par circle mein ghuma rahe ho. Ball chahti hai straight line mein ud jaaye — woh uska natural lazy path hai. Lekin string usse baar baar beech mein tumhare haath ki taraf kheenchti rehti hai. Woh constant inward kheench uske path ko circle mein mod deti hai. Inward bend ka woh "speed up" hi centripetal acceleration hai. Agar tum use zyada fast ghimate ho, tumhe bahut zyada kheeenchna padta hai — do guna fast ke liye chaar guna pull chahiye. Chod do, aur kuch bhi inward nahi kheench raha, toh ball seedhi shoot ho jaati hai.

Flashcards

Uniform circular motion mein ek object constant speed par bhi kyun accelerate karta hai?

Velocity ek vector hai; uski direction badhalti rehti hai, toh

Δ v \neq = 0

, jo non-zero acceleration deta hai jo center ki taraf directed hai.

Centripetal acceleration ke teen equivalent expressions kya hain?

a_{c} = v^{2} / r = ω^{2} r = v ω

Similar-triangles derivation mein, velocity vectors radius vectors ki tarah same angle

Δ θ

se kyun separate hain?

Kyunki har velocity apne radius ke perpendicular hai; dono vectors ko

90°

rotate karne se unke beech ka angle preserve hota hai.

∣Δ v ∣ = v Δ θ

kahan se aata hai?

Similar triangles se:

∣Δ v ∣/ v = ∣ AB ∣/ r = r Δ θ / r = Δ θ

Centripetal acceleration ki direction?

Center ki taraf, velocity ke perpendicular.

r = r (cos ω t, sin ω t)

use karke,

a

kya hai?

a = - ω^{2} r

, yaani magnitude

ω^{2} r

center ki taraf point karta hua.

Agar speed double ho aur radius fixed rahe, toh

a_{c}

kaise badlta hai?

Yeh chaar guna ho jaata hai, kyunki

a_{c} \propto v^{2}

Period

T

ke terms mein

a_{c}

express karo.

a_{c} = 4 π^{2} r / T^{2}

Kya centripetal force ek alag force hai?

Nahi — yeh net inward force hai, jo tension/gravity/friction/normal force se supply hoti hai.

Centripetal acceleration work kyun nahi karta?

Kyunki

a ⊥ v

, toh yeh direction badhalti hai speed nahi (aur force ⟂ displacement).

Connections

Newton's Second Law — $F_{c} = m a_{c}$ acceleration ko real inward forces se jodata hai.
Newton's First Law (Inertia) — kyun object warna straight jaata.
Angular velocity and period — $ω = 2 π / T$ aur $v = ω r$ ka source.
Banking of roads — application: normal force component as $F_{c}$ .
Gravitation — orbital motion — orbits ke liye gravity as centripetal force.
Vectors — derivative of a unit vector — calculus method ki backbone.
Centrifugal force (non-inertial frames) — "fictitious" counterpart.